High power density printed circuit board (PCB) embedded inductors

ABSTRACT

Devices, systems, and methods of manufacture relating to PCB embedded inductors are described in the present disclosure. Namely, an example device includes a substrate having an upper surface and an opposing lower surface. The device also includes a plurality of upper conductors disposed along the upper surface and a plurality of lower conductors disposed along the lower surface. The upper conductors and the lower conductors are radially disposed about a central axis. Each of the upper conductors and the lower conductors includes a petal shape. A distance between adjacent upper conductors is less than a width of each upper conductor and a distance between adjacent lower conductors is less than a width of each lower conductor. The device also includes a plurality of through-substrate conductors connecting respective upper conductors to respective lower conductors so as to form a series electrical connection. The series electrical connection includes a toroid configuration.

CROSS REFERENCE TO RELATED APPLICATION

The present disclosure claims priority to U.S. provisional patentapplication No. 62/259,146, filed on Nov. 24, 2015, the entire contentsof which are herein incorporated by reference.

BACKGROUND

Unless otherwise indicated herein, the materials described in thissection are not prior art to the claims in this application and are notadmitted to be prior art by inclusion in this section.

Direct current (DC)-to-alternating current (AC) power inverters mayinclude a passive filtering circuit. The filtering circuit may includean inductor and/or a capacitor (e.g., an LC circuit). Inductorsconfigured for switching operation below 1 MHz may utilize aferromagnetic or magnetic core. However, power inverters (e.g., GaNswitches) operating at higher switching frequencies (e.g., 5-10 MHz),may need smaller inductance values. As such, air-core inductors arefeasible for use in conventional power inverters. Among other features,when operated at high frequency, air-core inductors may offer lower costand higher efficiency as compared to other inductor configurations.Furthermore, printed circuit board (PCB) embedded air-core inductorsprovide the potential benefit of miniaturization and integration withpower electronic circuitry.

Various structures have been proposed for both embedded and non-embeddedair-core inductors. For example, solenoid, spiral, and toroidalconfigurations are the most common inductor structures. Inductors with atoroidal configuration may provide a potential benefit of reducedelectromagnetic interference (EMI) as compared to other alternativeconfigurations.

SUMMARY

The present disclosure relates to power inverter devices and systems. Inparticular, toroidal PCB-embedded air-core inductors are described. SomePCB-embedded inductors may have a toroidal configuration and may beconfigured to provide high frequency switching operation of powerinverter devices. Such inductors may be desirable in utility powerelectronics applications in order to provide improved efficiency,reduced size, better thermal conduction, higher parameter repeatability,and higher reliability at a reduced cost.

In a first aspect, a device is described. The device includes asubstrate having an upper surface and an opposing lower surface and aplurality of upper conductors disposed along the upper surface. Theplurality of upper conductors is radially disposed about a central axisand each upper conductor includes a petal shape. A distance betweenadjacent upper conductors is less than a width of each upper conductor.The device includes a plurality of lower conductors disposed along thelower surface. The plurality of lower conductors is radially disposedabout the central axis. Each lower conductor includes a petal shape. Adistance between adjacent lower conductors is less than a width of eachlower conductor. The device also includes a plurality ofthrough-substrate conductors connecting respective upper conductors torespective lower conductors so as to form a series electricalconnection. The series electrical connection comprises a toroidconfiguration.

In a second aspect, a system is described. The system includes a DCsource input and a switching circuit coupled to the DC source input. Thesystem also includes a filtering circuit coupled to the switchingcircuit. The filtering circuit includes an inductor. The inductorincludes a substrate having an upper surface and an opposing lowersurface and a plurality of upper conductors disposed along the uppersurface. The plurality of upper conductors is radially disposed about acentral axis. Each upper conductor comprises a petal shape. A distancebetween adjacent upper conductors is less than a width of each upperconductor. The system also includes a plurality of lower conductorsdisposed along the lower surface. The plurality of lower conductors isradially disposed about the central axis. Each lower conductor includesa petal shape. A distance between adjacent lower conductors is less thana width of each lower conductor. The system additionally includes aplurality of through-substrate conductors connecting respective upperconductors to respective lower conductors so as to form a serieselectrical connection. The series electrical connection includes atoroid configuration. The system yet further includes an output circuitcoupled to the filtering circuit.

In a third aspect, a method for designing a PCB-embedded inductor isdescribed. The method includes adjusting an outer diameter of theinductor and a number of turns of the inductor based on predeterminedphysical constraints and predetermined performance specifications. Theinductor includes a plurality of upper conductors disposed along anupper surface of a substrate. The plurality of upper conductors isradial disposed about a central axis. Each upper conductor has a petalshape. A trace clearance between adjacent upper conductors is less thana width of each upper conductor. The substrate includes a printedcircuit board (PCB). The inductor also includes a plurality of lowerconductors disposed along a lower surface of the substrate. The lowersubstrate is disposed opposite the upper surface. The plurality of lowerconductors is radially disposed about the central axis. Each lowerconductor includes a petal shape. A trace clearance between adjacentlower conductors is less than a width of each lower conductor. Theinductor also includes a plurality of through-substrate conductorsconnecting respective upper conductors to respective lower conductors soas to form a series electrical connection. The series electricalconnection includes a toroid configuration. The toroid configurationincludes a plurality of turns about a reference circle that is definedalong a reference plane between the upper surface and the lower surface.The method includes, subsequent to adjusting the outer diameter and thenumber of turns, determining whether an inductance of the inductor isgreater than a threshold inductance value. The method yet furtherincludes adjusting the trace clearance. The trace clearance comprises adistance between adjacent conductors. The method also includessubsequent to adjusting the trace clearance, determining whether aninductance of the inductor is greater than a threshold inductance value.The method includes adjusting a thickness of at least one of the upperconductor or the lower conductor. The method yet further includesadjusting a diameter of the through-substrate conductors. The methodadditionally includes adjusting a number of through-substrateconductors. The method also includes determining a plurality of deviceperformance metrics, based on the adjusted attributes of the inductor.The method also includes determining, based on the device performancemetrics being above a plurality of device requirements, that theinductor should be fabricated. The method also includes fabricating theinductor according to the adjusted attributes.

Other aspects, embodiments, and implementations will become apparent tothose of ordinary skill in the art by reading the following detaileddescription, with reference where appropriate to the accompanyingdrawings.

BRIEF DESCRIPTION OF THE FIGURES

FIG. 1A illustrates a device geometry.

FIG. 1B illustrates a device geometry.

FIG. 1C illustrates a device geometry.

FIG. 2A illustrates a filtering circuit, according to an exampleembodiment.

FIG. 2B illustrates an impedance versus frequency graph, according to anexample embodiment.

FIG. 2C illustrates an inductance versus capacitance graph, according toexample embodiments.

FIG. 2D illustrates an equivalent thermal circuit model, according toexample embodiments.

FIG. 3A illustrates a device, according to an example embodiment.

FIG. 3B illustrates a heat sink, according to an example embodiment.

FIG. 4A illustrates a device, according to an example embodiment.

FIG. 4B illustrates a device, according to an example embodiment.

FIG. 4C illustrates a device, according to an example embodiment.

FIG. 5 illustrates a resistance versus frequency graph, according toexample embodiments.

FIG. 6 illustrates a quality factor versus frequency graph, according toexample embodiments.

FIG. 7 illustrates an inductance versus frequency graph, according toexample embodiments.

FIG. 8 illustrates a thermal model and a heat map, according to exampleembodiments.

FIG. 9 illustrates a thermal model and a heat map, according to exampleembodiments.

FIG. 10 illustrates a thermal model and a heat map, according to exampleembodiments.

FIG. 11 illustrates a system, according to an example embodiment.

FIG. 12 illustrates a method of manufacture, according to an exampleembodiment.

FIG. 13 illustrates a method, according to an example embodiment.

FIG. 14 illustrates a three-dimensional graph of inductance versus ACresistance and number of turns, according to an example embodiment.

FIG. 15 illustrates a graph of inductance and efficiency versus numberof turns, according to an example embodiment.

DETAILED DESCRIPTION

Example methods, devices, and systems are described herein. It should beunderstood that the words “example” and “exemplary” are used herein tomean “serving as an example, instance, or illustration.” Any embodimentor feature described herein as being an “example” or “exemplary” is notnecessarily to be construed as preferred or advantageous over otherembodiments or features. Other embodiments can be utilized, and otherchanges can be made, without departing from the scope of the subjectmatter presented herein.

Thus, the example embodiments described herein are not meant to belimiting. Aspects of the present disclosure, as generally describedherein, and illustrated in the figures, can be arranged, substituted,combined, separated, and designed in a wide variety of differentconfigurations, all of which are contemplated herein.

As recited herein, the term “substantially” or “about” is meant todescribe that a characteristic, parameter, or value need not be achievedexactly, but that deviations or variations, including for example,tolerances, measurement error, measurement accuracy limitations andother factors known to skill in the art, may occur in amounts that donot preclude the effect the characteristic was intended to provide. Forexample, about 3000 mm² may include a range of areas with a tolerance of5% or within a range of 2850-3150 mm². As such, values expresslydescribed herein may also include reference to substantially similarvalues corresponding to tolerance ranges of 1%, 3%, 5%, 10%, or anothertolerance range.

Further, unless context suggests otherwise, the features illustrated ineach of the figures may be used in combination with one another. Thus,the figures should be generally viewed as component aspects of one ormore overall embodiments, with the understanding that not allillustrated features are necessary for each embodiment.

I. Overview

Devices, systems, and methods of manufacture relating to PCB embeddedinductors are described in the present disclosure. The PCB embeddedinductor includes a toroidal configuration and a dual-plane arrangementof metal layers that are electrically coupled by a plurality of throughsubstrate vias. The devices and systems described herein may providehigh efficiency (more than 98%) for high frequency inverterapplications. Such inverter devices and systems may be used for PVapplications. In an example embodiment, an inverter may include a ratedpower of 280 W with an input of 18 A and an output of 31.2V-DC voltage,9.07 A-DC current.

II. Example Devices

FIGS. 1A, 1B, and 1C illustrate several possible device geometries forinductors. FIG. 1A illustrates a first device geometry of an inductor.As illustrated in FIG. 1A, the inductor may include spiral configuration100. FIG. 1B illustrates a second possible device geometry of aninductor. As illustrated, the inductor may include solenoidconfiguration 110. FIG. 1C illustrates a third possible device geometry120 of an inductor. As illustrated, the inductor may include toroidconfiguration 120.

The spiral 100 and solenoid 110 configurations may providestraight-forward implementation with printed circuit board (PCB)technologies and may provide higher inductance values compared to thetoroid configuration 120. However, the toroid configuration 120 may beless susceptible to electromagnetic interference as compared to otherinductor configurations. For example, the toroidal configuration 120 mayprovide internal encapsulation of the magnetic field in the inductordevice.

Conventional switch mode power supplies (SMPSs) combine RF circuits andpower electronics operable at switching frequencies between 30-300 MHz(e.g., for use in resonant DC/DC converters). However, the presentdisclosure relates to toroidal PCB-embedded inductors suitable forsmoothing the AC current of hard-switched inverters operating within aswitching frequency range of about 1 to 10 MHz.

FIG. 2A illustrates a filtering circuit 200, according to an exampleembodiment. Such a filtering circuit 200 may be incorporated into ahard-switched inverter as described above. FIG. 2A may illustrate actualelectronic components (e.g., a resistor, a capacitor, and an inductor).Alternatively, FIG. 2A may illustrate an equivalent circuit (e.g., anequivalent resistance, an equivalent capacitance, and an equivalentinductor. Parasitic capacitance, inductance, and resistance values mayalso be represented and/or accounted for in such a circuit.

FIG. 2B illustrates an impedance versus frequency graph 210, accordingto an example embodiment. The graph 210 illustrates behavior of anexample filtering circuit over a wide range of frequencies using aspectrum analyzer. The example filtering circuit may have acharacteristic resonance at approximately 280 MHz. However, otherresonant frequencies are possible. A resonant frequency of the filteringcircuit may be based, at least in part, on the inductance and thecapacitance of conductors and vias.

To design a filtering circuit (e.g., filtering circuit 200) to smooththe AC current of a hard-switched inverter, a cutoff frequency of thefiltering circuit may be selected. In some embodiments, the cutofffrequency may be approximately 1/10 the switching frequency of theinverter. For a parallel LC circuit,

$f_{cutoff} = {\frac{1}{2\pi\sqrt{L_{m\; i\; n}C_{m\; i\; n}}} = \frac{1}{\sqrt{\frac{4{\pi^{2}\left( {V_{i\; n} - V_{out}} \right)}D_{{ma}\; x}}{\Delta\; I \times f_{sw}}} \times \frac{\Delta\;{ID}_{{ma}\; x}}{f_{sw}\Delta\; V}}}$

where V_(in) is the input voltage, V_(out) is the output voltage,D_(max) is the maximum duty cycle of an input signal, f_(sw) is theswitching frequency of the inverter, and f_(cutoff) is the cutofffrequency of the LC circuit. Accordingly, for a given switchingfrequency, an appropriate value for the inductance of a LC circuit maybe expressed as:

$L = \frac{\left( {V_{i\; n} - V_{out}} \right)D_{{ma}\; x}^{2}}{\Delta\; V \times f_{sw}^{2} \times C}$

FIG. 2C illustrates an inductance versus capacitance graph 220,according to example embodiments. Namely, FIG. 2C illustratescorresponding inductance and capacitance values appropriate forfiltering circuits in full bridge inverter circuits operating at 5 MHzand 10 MHz switching frequencies. Within the context of the presentdisclosure, FIG. 2C may provide design parameters for components of thefiltering circuits, including appropriate inductance values for thedesign of inductor devices described herein. In example embodiments, inorder to reduce or eliminate the need for large capacitance values (andcorrespondingly large passive devices) in the filtering circuit,inductance values between 30-500 nanoHenries (nH) may be considered. Insome embodiments, a low-loss ceramic capacitor may be provided in thefilter circuit. In such scenarios, the capacitor may have a capacitancevalue of 1-2 μF.

FIG. 2D illustrates an equivalent thermal circuit model 230, accordingto example embodiments. The equivalent thermal circuit model 230 mayinclude a thermal resistivity of copper (R_(Copper)), the thermalresistivity of the printed circuit board (R_(PCB)), and the thermalresistivity of air (R_(Air)). As described elsewhere herein, temperaturerise under normal operating conditions may be quantified using thermalcircuit model 230.

FIG. 3A illustrates a device 300, according to an example embodiment.Device 300 includes a substrate 310 (illustrated herein as beingtransparent for clarity) having an upper surface and an opposing lowersurface. For example, the substrate 310 could include a printed circuitboard material, such as FR-4 or another type of resin-epoxy compositematerial. The upper surface of the substrate 310 could be a top (outersurface) of the substrate 310. Alternatively, the upper surface of thesubstrate 310 could be a layer of substrate 310 that is positioned abovethe lower surface. Likewise, the lower surface could be a bottom surfaceof the substrate 310 or a layer of the substrate 310 that is positionedbelow the upper surface. The device 300 may be formed using a two-layercopper PCB. Other types of substrate materials are possible.

The device 300 also includes a plurality of upper conductors 320. Theupper conductors 320 are disposed along the upper surface of thesubstrate 310. In some embodiments, the upper conductors 320 may bedisposed on a top surface of a printed circuit board. The plurality ofupper conductors 320 is radially disposed about a central axis. That is,as illustrated in FIG. 3A, the central axis may pass through an openmiddle portion of the device 300. In such a scenario, the upperconductors 320 may be arranged in a rotationally-symmetric arrangement.Additionally or alternatively, the upper conductors 320 may be arrangedabout the central axis at a minimum radius setback. That is, an inneredge of the respective upper conductor 320 may be disposed along theupper surface at a distance away from the central axis that correspondsto the minimum radius setback. As a further alternative, the upperconductors 320 may be arranged such that an outer edge of the respectiveupper conductor 320 may be disposed at a maximum radius setback from thecentral axis.

Each upper conductor 320 may be shaped in a petal shape. For example,each upper conductor 320 may have an inner edge located closest to thecentral axis and an outer edge located away from the central axis. Insuch a scenario, the inner edge could be narrower than the outer edge.Put another way, the petal shape could be indicative of a width along anouter perimeter of a given upper conductor 320 that is larger than awidth along an inner perimeter of the given upper conductor 320. Yetfurther, the petal shape could include the plurality of upper conductors320 appearing like petals of a blooming flower (e.g., a daisy orsunflower). Furthermore, a distance between adjacent upper conductorsmay be less than a width of each upper conductor 320. That is, the upperconductors 320 could be shaped, arranged, and/or spaced such that aspace between adjacent upper conductors 320 may be narrower compared tothe width of each upper conductor 320. It will be understood that insome scenarios, the upper conductors 320 need not be shaped identicallyand that minor deviations in shape are possible between the respectiveupper conductors 320.

The device 300 also includes a plurality of lower conductors 322. Thelower conductors 322 are disposed along the lower surface of thesubstrate 310. For example, in some embodiments, the lower surface ofthe substrate 310 may include a bottom surface of a printed circuitboard. The plurality of lower conductors 322 is radially disposed aboutthe central axis. In some embodiments, the plurality of lower conductors322 may be arranged such that they are aligned with portions of theupper conductors 320. That is, the plurality of lower conductors 322 maybe disposed such that an inner portion of a given upper conductor 320overlaps an inner portion of a first lower conductor 322 while an outerportion of the given upper conductor 320 overlaps an outer portion of asecond lower conductor 322.

Each lower conductor 322 may be shaped in a petal shape. For example,each lower conductor 322 may have an inner edge located closest to thecentral axis and an outer edge located away from the central axis. Insuch a scenario, the inner edge could be narrower than the outer edge.As described with regard to the upper conductors 320, the petal shapecould be indicative of a width along an outer perimeter of a given lowerconductor 322 that is larger than a width along an inner perimeter ofthe given lower conductor 322. Furthermore, the petal shape couldinclude the plurality of lower conductors 322 appearing like petals of ablooming flower (e.g., a daisy or sunflower). Additionally, a distancebetween adjacent lower conductors may be less than a width of each lowerconductor 322. That is, the lower conductors 322 could be shaped,arranged, and/or spaced such that a space between adjacent lowerconductors 322 may be less than the width of each lower conductor 322.

The device 300 also include a plurality of through-substrate conductors330 and 332 connecting respective upper conductors to respective lowerconductors so as to form a series electrical connection. Thethrough-substrate conductors 330 and 332 may be conductive viasconfigured to electrically couple a given upper conductor to a lowerconductor. In some embodiments, each upper conductor may include one ormore inner through-substrate conductors 332 and one or more outerthrough-substrate conductors 330. As illustrated in FIG. 3A, a givenlower conductor (e.g., lower conductor 332 a) may be connected to twodifferent upper conductors so as to form a series electrical connection.That is, upper and lower conductors may be connected in a seriesclockwise or counter-clockwise arrangement. In such a manner, the serieselectrical connection may include a toroid configuration.

Put another way, the toroid configuration of device 300 may include aplurality of turns of a conductor about a reference circle 350 that isdefined along a reference plane between the upper surface and the lowersurface. As such, device 300 may represent a way to form a toroidinductor (e.g., toroid 120) using printed circuit board technology.

The device 300 may include an input leg 340 and an output leg 342.Current may enter device 300 via the input leg 340 be circulated throughthe through-substrate conductors and petal conductors (e.g., theplurality of upper conductors and lower conductors) and exit the device300 via the output leg 342. It will be understood that input leg 340 andoutput leg 342 may additionally or alternatively be connected to an ACsignal, in which case, current may flow in alternating directions withinthe device 300.

The inductance of a toroidal PCB embedded inductor may be calculated by:

${L = {{\frac{N^{2}h\;\mu_{0}}{2\pi}{\ln\left( \frac{d_{o}}{d_{i}} \right)}} + {\frac{d_{o} + d_{i}}{4}{\mu_{0}\left\lbrack {{\ln\left( {8\frac{d_{o} + d_{i}}{d_{o} - d_{i}}} \right)} - 2} \right\rbrack}}}},$

wherein N is the number of turns, h is the height of the inductor (e.g.,the PCB thickness), and μ₀ is the permeability of air. Furthermore,d_(i) and d_(o) are the inner and outer diameters of the toroidalinductor, respectively.

In some embodiments, the inductance may be expressed in the simplifiedform:L∝μ ₀ kN ² h,

where k is a constant value for defined outer/inner inductor geometries.

Although miniaturizing inductor components may provide higher powerdensity, increased resistance may provide unwanted effects such ascopper conduction losses. The resistance of device 300 may include twoparts: resistance of the petals (e.g., the upper and lower conductors)and the resistance of the vias (e.g., the through-substrate conductors).In an example embodiment, the resistance of device 300 may be expressedas:

${R = {N\left( {{2 \times R_{petal}} + \frac{R_{{inner}\;\_\;{via}}}{N_{inner}} + \frac{R_{{outer}\;\_\;{via}}}{N_{outer}}} \right)}},$

The resistance of each petal is proportional to the copper resistivity,number of turns, and outer diameter of the inductor, and it is inverselyrelated to the skin depth, petals' clearance (e.g., the distance betweenadjacent conductors) and inner diameter of the vias. Via resistance willscale directly with copper resistivity and inversely with the viadiameter and the skin depth, which is a measure of the penetration of aplane electromagnetic wave into the copper. By considering the impact ofskin and proximity effects on the total resistance of the vias, thecurrent tends to pass through the edges of the conductor (copper) as thecurrent density is largest near the surface. Also, proximity effect dueto the relatively bigger size of each petal with respect to theirclearance can be ignored. In an embodiment, the resistance of device 300with an alternating signal may be expressed as:

${R_{a\; c} = \frac{N^{2}}{G_{0} - {{N\left( {\beta - 1} \right)}\delta\; l}}},$

where G₀ is the global conductivity, 1 is the average length of thecurrent path, and δ is the skin depth of the conductive material (e.g.,copper) in the inductor. By simplifying the equation and applying theskin effect relation with switching frequency, the AC resistance can beexpressed as:R _(ac) ∝k ₁ N ² √{square root over (f)},

where k₁ is a constant factor that depends on the geometry of theinductor.

In some embodiments, JMAG finite element analysis (FEA) simulations maybe conducted to predict the thermal and electrical properties forvarious inductor designs. The various inductor designs may includevarying numbers and size of the through-substrate conductors.

In some embodiments, a thickness of the substrate 310 may be less thanabout 1.6 millimeters. In other embodiments, the thickness of thesubstrate 310 may be in the range of 0.5 millimeters to 2 millimeters.Alternatively, the thickness of substrate 310 may be another value.

In an example embodiment, a diameter of the device 300 may be less thanabout 60 millimeters. For instance, an outer diameter of device 300(e.g., corresponding to an outer edge of the upper and/or lowerconductors) may be approximately 30 millimeters. It will be understoodthat other values for the diameter of device 300 may be possible.

In example embodiments, an area of the device 300 may be less than about3000 mm². For example, the area of device 300 may be approximately 900mm². It is understood that other values for the area of device 300 arepossible and contemplated herein.

In some embodiments, device 300 may include more or fewer turns. Forexample, the plurality of turns may be 8, 13, or 26. In some examples,the plurality of turns may be less than 50, and more than 4. Otherexamples are possible as well. In some cases, the plurality of turns maydirectly affect the inductance of device 300. That is, all otherparameters being equal, as the number of turns increases, the inductanceof device 300 will increase. For example, as described herein, an 8-turninductor device may have an inductance of 50 nH and a 26-turn inductordevice may have an inductance of 242 nH. As such, the number of upperconductors and/or the number of lower conductors may vary based on, forexample, a desired inductance value.

Inductance values of devices disclosed herein may be between about 45and 305 nanoHenries. In an example embodiment, a desired inductancevalue may be approximately 50 nH. However, other values of inductanceare possible and contemplated.

In some embodiments, a resistance of the device 300 may be less thanabout 2.6 milliOhms at DC and less than about 9 milliOhms for 1 MHzsignal. For example, in the case of an 8-turn inductor device, theresistance may be approximately 2.4 mOhms. Resistance values will varybased on, among other aspects, the number of turns of device 300.

In an example embodiment, a quality factor of the device 300 may be atleast 300, however other values for the quality factor are possible.

As described throughout the present disclosure, three inductor designswere modeled, built and tested, as described in Table I, to illustratevarious parameters and configurations of the device that result in thoseparameters.

TABLE I Physical Parameters of PCB Inductor Designs Parameters Design 1Design 2 Design 3 Units Inductance 242 150 50 nH Number of Turns 26 13 8— Outer Diameter 60 60 30 mm Inner Diameter 20 20 15 mm DC Resistance160 32 2.4 mΩ AC Resistance 200 50 8 mΩ Via Diameter 1 2 2.8 mm Height1.5 3.5 3.5 mm

TABLE II Physical Parameters of PCB Inductor Designs at 1 MHzFrequency-1 MHz Design1 Design2 Design 3 Units 1-Oz 149.8 29.5 5.2 mΩ2-Oz 69.5 15 2.7 mΩ 4-Oz 39.5 7.8 3 mΩ 6-Oz 27.9 5.6 2.1 mΩ

TABLE III Physical Parameters of PCB Inductor Designs at 10 MHzFrequency-10 MHz Design1 Design2 Design 3 Units 1-Oz 208.5 57.6 26.7 mΩ2-Oz 108 28.1 14.5 mΩ 4-Oz 60.1 14.6 7.9 mΩ 6-Oz 46.1 11 5.9 mΩ

Tables II and III illustrate the effects of copper conductor thicknesson AC resistance for each of the three designs at 1 MHz and 10 MHz,respectively.

FIG. 3B illustrates a heat sink 360, according to an example embodiment.In some embodiments, device 300 may also include a heat sink 360 coupledto at least one of the upper surface or at least one of the upperconductors 320. The heat sink 360 may be coupled to the upper surface orthe upper conductors 320 by a base portion 366. For example, the baseportion 366 may be clamped, glued, or otherwise fastened to the uppersurface or the upper conductors 320. Furthermore, when coupled, the baseportion 366 may include thermal paste or another thermally conductivematerial so as to efficiently conduct heat away from the upper surfaceand/or other portions of device 300. The heat sink 360 may includecooling fins 362 configured to dissipate heat over a fin area 364. Theheat sink 360 may dissipate heat via, for example, thermal conductionand thermal convection.

The total thermal resistance of a device 300 and a heat sink 360 may beexpressed as:

${R_{{th}\;\_\;{heatsink}} = \frac{1}{h\left( {A_{heatsink} + {\eta\; N_{fin}A_{fin}}} \right)}},$

where h is the convective heat transfer coefficient, A_(heatsink) is thearea of the base portion 366, A_(fin) is the surface area of the fins(e.g., fin area 364), and N_(fin) is the total number of cooling fins362 in the heat sink 360. The total thermal resistivity of thePCB-embedded inductor may be expressed as:R _(th) _(_) _(total) =R _(th) _(_) _(heatsink)+(R _(th) _(_) _(PCB) ∥R_(th) _(_) _(copper)), where

$R_{{th}\;\_\;{PCB}} = \frac{1}{h_{PCB}A_{inductor}}$

According to the above equations, the addition of a heat sink in device300 may provide some design and performance benefits. For example,keeping a temperature rise value constant (e.g., 45° C.), an inductordiameter may be reduced from 60 mm to 39 mm for the 26-turn and the13-turn inductor designs, respectively. Furthermore, again keepingtemperature rise constant, the inductor diameter may be decreased from30 mm to 25 mm for the 8-turn inductor design. As a result of decreasingthe inductor diameter, the total resistance of the PCB-embedded inductorwill be decreased by 20% for the 26-turn and 13-turn inductors and 10%for the 8-turn inductor, resulting in higher efficiency.

FIG. 4A illustrates a device 400, according to an example embodiment.Device 400 illustrates a variation of device 300 that includes 8-turns(e.g., Design 3 from Table I). That is, the series electrical connectionof device 400 includes a toroid configuration with 8 turns about thereference circle 350. Device 400 may have a diameter of approximately 30mm.

FIG. 4B illustrates a device 430, according to an example embodiment.Device 430 illustrates a variation of device 300 that includes 13-turns(e.g., Design 2 from Table I). That is, the series electrical connectionof device 430 includes a toroid configuration with 13 turns about thereference circle 350. Device 430 may have a diameter of approximately 60mm.

FIG. 4C illustrates a device 450, according to an example embodiment.Device 450 illustrates a variation of device 300 that includes 26-turns(e.g., Design 1 from Table I). That is, the series electrical connectionof device 450 includes a toroid configuration with 26 turns about thereference circle 350. Device 450 may have a diameter of approximately 60mm.

FIGS. 5, 6, and 7 include results from several finite element analysissimulations performed on each of the PCB-embedded inductor designsdescribed in Table I. FIG. 5 illustrates a resistance versus frequencygraph 500, according to example embodiments. Namely, FIG. 5 illustratesthe resistance of 8-turn, 13-turn, and 26-turn designs with respect tosignals with frequencies between 1 to 10 MHz, based on finite elementanalysis simulations.

FIG. 6 illustrates a quality factor versus frequency graph 600,according to example embodiments. Specifically, FIG. 6 illustrates thequality factor of 8-turn, 13-turn, and 26-turn designs with respect tosignals with frequencies between 1 to 10 MHz, based on finite elementanalysis simulations. The conduction loss and filtering quality ofdevice 300 may be based, at least in part, to the quality factor. Assuch, in some embodiments, various aspects of the inductor (e.g., copperthickness, via diameter, upper and lower conductor arrangements, numberof turns) may be selected so as to maximize the quality factor.

FIG. 7 illustrates an inductance versus frequency graph 700, accordingto example embodiments. Namely, FIG. 7 illustrates the inductance of8-turn, 13-turn, and 26-turn designs with respect to signals withfrequencies between 1 to 10 MHz, based on finite element analysissimulations.

Under steady-state operation and assuming convection effects, atemperature rise of devices and systems described herein may be a resultof power dissipated in the inductor. In example embodiments, thetemperature rise may be approximated by:ΔT=P×R _(th),

where P is the total dissipative power of the inductor and R_(th) is thetotal thermal resistivity between the substrate and the environment. Inanother form, the dissipated power may be expressed as:P=UAΔT,

where P is the total dissipative power of the inductor, U is the globalthermal coefficient, and A is the global surface area for thePCB-embedded inductor. Accordingly, given a constant quality factor,either inductance or resistance (or both) may be reduced to decrease theamount of dissipative power. Normalizing by quality factor, thetemperature rise of the PCB-embedded inductor may be expressed as:ΔT _(P.U)=^(P×R) ^(th) /Q,

where ΔT_(P.U) is the temperature rise per quality factor increment.

The quality factor may be calculated based on the expression:

$Q = {\frac{\omega\; L}{R_{a\; c}} = {\frac{2\pi\;{fL}}{R_{a\; c}} = {\frac{k_{1}2\pi\;{fxN}^{2}}{k_{3}N^{2}\sqrt{f}} = {k_{0}x{\sqrt{f}.}}}}}$

Accordingly, the temperature rise may be expressed as:

$\begin{matrix}{{{\Delta\; T_{P \cdot U}} = {{\left( \frac{P}{UA} \right)/Q} = {{k\frac{I^{2}Z}{x^{2} \times x\sqrt{f}}} = {k\frac{I^{2}Z}{x^{3}\sqrt{f}}}}}},} \\{{= {{k\frac{I^{2}\sqrt{\left( {2\pi\;{fL}} \right)^{2} + R_{a\; c}^{2}}}{x^{3}\sqrt{f}}} \propto {k\frac{I^{2}\sqrt{\left( {2\pi\;{fL}} \right)^{2} + f}}{x^{3}\sqrt{f}}}}},{and}} \\{{= {k_{0}\frac{I^{2}\left( {\sqrt{f} + \alpha_{0}} \right)}{x^{3}}}},}\end{matrix}$

where x is the ratio of surface area to an effective length of theinductor, Z is the impedance, I is the root mean square current throughthe inductor, and Q is the quality factor of the inductor. ΔT_(P,U) isthe temperature rise per unit of the quality factor of the inductor, k₀is a constant based on the geometry of the device, and α₀ is an ACresistance variation of the inductor. The variable x is defined as adegree of freedom for a design that includes an area of the inductordivided by an outer diameter of the inductor. A design with a largervalue for x, which corresponds to a bigger surface area to diameter, mayprovide better heat transfer capability. However, a larger value for xmay reduce power density and/or device efficiency.

FIGS. 8, 9, and 10 illustrate thermal models and actual temperatureprofiles of several example devices described herein. With theabove-described design considerations, several different PCB-embeddedinductors were simulated through using finite element methods in JMAG.That is, the thermal models 800, 900, and 1000 were determined, at leastin part, by finite element analysis software. The thermal models 800,900, and 1000 may be determined additionally or alternatively based onthe equations described above. Among other information, the finiteelement simulations verified the predicted results for temperaturedistribution, current density, AC resistance, and inductance of thePCB-embedded inductor designs.

Furthermore, prototypes of each of the PCB-embedded designs were testedcontinuously under nominal current values (5 A for 26-turns, 10 A for13-turn, and 18 A for 8-turn) for 20 minutes to reach the steady stateconditions to extract the temperature rise profile (e.g., heat map 820,920, and 1020).

FIG. 8 illustrates a thermal model 800, corresponding temperature index810, and a heat map 820, according to example embodiments. Namely, FIG.8 illustrates respective temperature profiles of simulation andexperimental results of a 26-turn design. The design included an outerdiameter of 60 mm, copper thickness of 35 μm, via diameter of 2 mm,inductance of 242 nH, a DC resistance of 160 mΩ, and an AC resistance at5 MHz of 200 mΩ. The applied current was 5 A, and the ambienttemperature was 23° C. Temperature rise experiments based on the 26-turndesign showed temperature rise greater than 35° C. for 5 A appliedcontinuous current.

FIG. 9 illustrates a thermal model 900, corresponding temperature index910, and a heat map 920, according to example embodiments. Namely, FIG.9 illustrates respective temperature profiles of simulation andexperimental results of a 13-turn design. The design included an outerdiameter of 60 mm, copper thickness of 35 μm, via diameter of 2 mm,inductance of 150 nH, a DC resistance of 32 mΩ, an AC resistance at 5MHz of 50 mΩ. The applied current was 12 A, and the ambient temperaturewas 23° C. Temperature rise experiments based on the 13-turn designshowed temperature rise greater than 40° C. for 12 A applied continuouscurrent.

FIG. 10 illustrates a thermal model 1000, corresponding temperatureindex 1010, and a heat map 1020, according to example embodiments.Namely, FIG. 10 illustrates respective temperature profiles ofsimulation and experimental results of an 8-turn design. The designincluded an outer diameter of 30 mm, copper thickness of 144 μm, viadiameter of 2.8 mm, inductance of 50 nH, a DC resistance of 2.4 mΩ, anAC resistance at 5 MHz of 8 mΩ. The applied current was 18 A, and theambient temperature was 23° C.

Simulation and experimental results were utilized to verify other designchoices. For example, simulations and experiments verified that thetemperature reaches the highest point where the output leg is bent, andthe current path is diverted in a different direction. Additionally, thehighest attainable temperature is expected and verified in the 26-turninductor design, which has higher conductive losses through the petals.The higher conductive losses result in the higher temperature rise.

III. Example Systems

Inductor devices described herein may be incorporated into varioussystems. Such systems may include, for example, DC/AC inverter systems,switch mode power supplies (SMPSs), or hard switched inverters. FIG. 11illustrates a system 1100, according to an example embodiment. System1100 may include a DC source input 1110. In an example embodiment, aninput current of the DC source input 1110 may be at least 17 Amperes.The DC source input 1110 may include other values of DC voltage and DCcurrent.

The system 1100 includes a switching circuit 1120 coupled to the DCsource input 1110. In an example embodiment, the switching circuit 1120may include a gallium nitride (GaN) transistor-based half-bridgeinverter. In such a scenario, a switching frequency of the switchingcircuit 1120 may be between about 1 MHz and 30 MHz. Other types ofswitching circuits and switching frequencies are contemplated. Forexample, full-bridge inverters, converters, other power circuits, andcascaded versions thereof are contemplated herein.

As an example, the switching circuit 1120 may include an Efficient PowerConversion EPC9033 development board. For example, the EPC9033 mayinclude enhancement mode GaN field effect transistors (FETs) that may beoperated in a half-bridge inverter mode. In some cases, the switchingcircuit 1120 may use a pulse-wave modulated (PWM) signal as an inputsignal for switching the DC voltage and current signal from the DCsource input 1110.

The system 1100 also includes a filtering circuit 1130 coupled to theswitching circuit 1120. In an example embodiment, the switching circuit1120 provides a pulse-wave modulated (PWM) signal and/or a square wavesignal to the filtering circuit 1130. That is, the switching circuit1120 may switch a signal from the DC source input 1110 so as to providea PWM signal or square wave signal based on the desired switchingfrequency.

In an example embodiment, the filtering circuit 1130 may be configuredto filter out low frequency ripple currents and high frequency harmonicsin signals provided by the switching circuit 1120. The filtering circuit1130 includes an inductor 1132, and optionally, a capacitor. As providedelsewhere herein, the filtering circuit 1130 may include a parallel LCcircuit with a resonant frequency that may be approximated by:

${f_{cutoff} = {\frac{1}{2\pi\sqrt{L_{m\; i\; n}C_{m\; i\; n}}} = \frac{1}{\sqrt{\frac{4{\pi^{2}\left( {V_{i\; n} - V_{out}} \right)}D_{{ma}\; x}}{\Delta\; I \times f_{sw}}} \times \frac{\Delta\;{ID}_{{ma}\; x}}{f_{sw}\Delta\; V}}}},$

where D_(max) is the maximum duty cycle of an input signal, f_(sw) isthe switching frequency of the switching circuit 1120, and f_(cutoff) isthe cutoff frequency of the AC filter. In some embodiments, f_(cutoff)may be set to less than 1/10 f_(sw). In such a scenario, the desiredinductance value of inductor 1132 in the filtering circuit 1130 may beapproximated by:

$L = {\frac{\left( {V_{i\; n} - V_{out}} \right)D_{{ma}\; x}^{2}}{\Delta\; V \times f_{sw}^{2} \times C}.}$

The inductor 1132, which may be similar or identical to device 300 asillustrated and described in relation to FIG. 3A, includes a substratehaving an upper surface and an opposing lower surface. The inductor alsoincludes a plurality of upper conductors disposed along the uppersurface of the substrate. The plurality of upper conductors is radiallydisposed about a central axis. Each upper conductor has a petal shapeand a distance between adjacent upper conductors is less than a width ofeach upper conductor.

Furthermore, the inductor 1132 includes a plurality of lower conductorsdisposed along the lower surface. The plurality of lower conductors isradially disposed about the central axis. Each lower conductor has apetal shape and a distance between adjacent lower conductors is lessthan a width of each lower conductor.

The inductor 1132 further includes a plurality of through-substrateconductors connecting respective upper conductors to respective lowerconductors so as to form a series electrical connection. The serieselectrical connection includes a toroid configuration.

The system 1100 further includes an output circuit 1140 coupled to thefiltering circuit 1130. The output circuit 1140 may include an outputamplifier and/or output contacts.

Optionally, the system 1100 may include a controller 1150. Thecontroller 1150 includes at least one processor 1152 and a memory 1154.The at least one processor 1152 may carry out instructions stored in thememory 1154 so as to carry out operations. The operations may include,but are not limited to, causing the switching circuit 1120 to modulate asignal from the DC source input 1110 based on a desired switchingfrequency.

In an example embodiment, the desired switching frequency may be between1-10 MHz. However, other switching frequencies are possible andcontemplated herein.

Other operations are contemplated. Namely, the controller 1150 may beconfigured to control and/or adjust any or all of DC source input 1110,the filtering circuit 1130, or the output circuit 1140.

In an example embodiment, the system 1100 may be operated in acontinuous conduction mode. That is, in such a scenario, a peak outputvoltage of the output circuit 1140 is at least 30 Volts and an outputcurrent of the output circuit 1140 may be at least 9 Amperes. It will beunderstood that other values for the peak output voltage and the outputcurrent of the output circuit 1140 are possible and contemplated herein.

While continuous conduction mode operation is described above, thesystem 1100 may be additionally or alternatively operated in adiscontinuous conduction mode.

In an example embodiment, during operation of the system 1100, atemperature rise of the inductor 1132 may be no greater than about 45°C. While other temperature rise values are possible, the design ofinductor 1132 may be selected so as to minimize temperature rise and/orheat distribution within the inductor 1132.

In some embodiments, a total harmonic distortion (THD) of the outputcircuit 1140 is less than about 10%. Namely, after filtering, the THDfor the 8-turn design may be about 9.5%, the THD for the 13-turn designmay be about 6.8%, and the THD for the 23-turn design may be about 5.8%.Other THD values are possible. Furthermore, various aspects of inductor1132 and the filtering circuit 1130 may be selected based on the THD ofsystem 1100 and output circuit 1140. Namely, the resonance frequency,frequency cutoff, or bandwidth of filtering circuit 1130 may be selectedso as to minimize or reduce the THD of the output circuit 1140.

In some example embodiments, system 1100 may include one or more stagesof a Si-based AC-stacked inverter. In some embodiments, the inductor1132 may be a PCB-embedded inductor with a toroidal configuration. Suchan inductor may operate at relatively high efficiency (e.g., more than98%) at switching frequencies greater than 1 MHz. Assuming approximately2-5% power loss for the switching circuit, including conduction andswitching losses, the maximum allowable loss for the AC filter shouldgenerally be lower than 1-2% to achieve more than 95% overall efficiencyfor the inverter system. Under such conditions, system 1100 may have arated power of 280 W. Specifically, the DC source input may be greaterthan 30 Volts with a 9 Amp input current. In such a scenario, theinverter output current may be 18 Amperes or more.

IV. Example Methods

FIG. 12 illustrates a method of manufacture 1200, according to anexample embodiment. The method 1200 includes blocks that may be carriedout in any order. Furthermore, various blocks may be added to orsubtracted from method 1200 within the intended scope of thisdisclosure. The method 1200 may correspond to steps that may be carriedout using any or all of the devices and systems illustrated anddescribed in reference to FIG. 3, 4A, 4B, 4C, or 11. That is, asdescribed herein, method of manufacture 1200 may relate to forming aninductor device, which may be an element in an inverter system.

Block 1202 includes forming a plurality of upper conductors along anupper surface of a substrate. As described elsewhere herein, thesubstrate may include a printed circuit board material, such as FR-4. Insuch a scenario, the plurality of upper conductors is radially disposedabout a central axis. That is, the upper conductors may be disposed in aradial pattern about the central axis.

Each upper conductor may be formed in a petal shape (e.g., wider alongan outer edge as compared to an inner edge. A distance between adjacentupper conductors may be less than a width of each upper conductor.

Block 1204 includes forming a plurality of lower conductors disposedalong a lower surface of the substrate. The lower substrate is disposedopposite the upper surface. For example, the lower substrate may be abottom surface of the FR-4 material and the upper surface may be a topsurface of the FR-4. The plurality of lower conductors could be radiallydisposed about the central axis and each lower conductor could include apetal shape. In such a scenario, a distance between adjacent lowerconductors is less than a width of each lower conductor.

In an example embodiment, forming the upper conductors and/or the lowerconductors may include standard PCB processing techniques, such asexposing and developing a resist mask on the PCB and using electrolessmetal plating to form the copper conductors in the unmasked regions.Additionally or alternatively, the substrate may be patterned with othertypes of additive and/or subtractive techniques. For example, theconductors may be deposited using metal evaporation, electrolysis, orother electroplating methods. The conductors may be removed, at least inpart, using etching and/or liftoff techniques. Other semiconductormanufacturing and PCB processing techniques to form the upper conductorsand/or the lower conductors are contemplated herein.

Block 1206 includes forming a plurality of through-substrate conductorsthat connect respective upper conductors to respective lower conductorsso as to form a series electrical connection. The series electricalconnection may include a toroid configuration. In such a scenario, thetoroid configuration may include a plurality of turns about a referencecircle that is defined along a reference plane between the upper surfaceand the lower surface. For example, the plurality of turns may bebetween 8 and 26.

In some embodiments, forming the plurality of through-substrateconductors may include drilling or punching the substrate to form a viaand then plating along the sidewalls of the via to create an electricalconnection between the upper conductors and the lower conductors. Inother embodiments, a conductive plug may be formed in the via. Otherways of electrically connecting the respective upper conductors with therespective lower conductor in a serial connection so as to form atoroidal configuration are contemplated herein.

Optionally, the method may include forming a heat sink that is coupledto at least one of the upper surface or the upper conductor. Forexample, the heat sink may include a plurality of cooling finsconfigured to remove heat from the inductor device. In some embodiments,the upper surface and the heat sink may be coated, at least in part,with a thermal paste or a thermal compound, which may have a highcoefficient of thermal conductivity.

FIG. 13 illustrates a method 1300, according to an example embodiment.The method 1300 includes blocks that may be carried out in any order.Furthermore, various blocks may be added to or subtracted from method1300 within the intended scope of this disclosure. The method 1300 maycorrespond to steps that may relate to the design and simulation of thedevices and systems illustrated and described in reference to FIG. 3,4A, 4B, 4C, or 11. That is, as described herein, method 1300 may relateto a design optimization of a PCB-embedded inductor device, which may beutilized in power electronic device filter. In some embodiments, method1300 may provide design and optimization of a high power densityinductor to maximize its efficiency when applied to inverterapplications. In some embodiments, method 1300 may be conducted with oneor more blocks of method 1200 as illustrated and described in referenceto FIG. 12.

Method 1300 may represent a multi-objective algorithm that may helpprovide design solutions with regard to various design goals (e.g.,device current, geometries, inductance, resistance, etc.). In exampleembodiments, method 1300 may include an iterative algorithm thatincludes 6 different levels (e.g., level-0 to level-5). Level-0 includesconsideration of the electrical and geometrical design specificationsand constraints.

Block 1302 includes determining and/or imposing certain designspecifications for a desired inductor device. For example, the designspecifications may include inductance value (e.g., 50-300 nH), maximumtemperature rise, power efficiency, voltage, AC resistance, switchingfrequency, and the RMS current of the inductor. Such designspecifications may be used initially, or throughout one or more blocksof method 1300.

Block 1304 includes determining and/or imposing certain physicalconstraints for the desired inductor device. As an example, the physicalconstraints may include a number of turns, an inner diameter, an outerdiameter (e.g., not more than 60 mm), an overall device footprint, etc.In some embodiments, the inner diameter (e.g., 10 mm or 15 mm) and theouter diameter may be fixed (e.g., 30 or 60 mm). In such scenarios, thenumber of turns largely defines the inductance of the desired inductordevice. Increasing the number of turns in the inductor device mayprovide better filtering characteristics and the inductance may scale asL∝N². However, AC resistance and conduction loss will increase with anincreased number of turns, which may lead to higher temperature rise andlower quality factor. As the inner diameter decreases, inductanceincreases. However, increasing the number of turns in the inductor alsoincreases the device resistance, which may reduce efficiency of thedevice. Yet further, higher resistance increases the device temperaturerise due to higher power dissipation in the inductor.

Block 1306 (level-1) includes increasing the outer diameter from aninitial value (e.g., 10 mm).

Block 1308 includes updating the minimum fabrication constraints (e.g.,substrate size, thickness, etc.) based on the outer diameter value fromblock 1306.

Block 1310 (level-2) includes increasing a number of turns from aninitial value (e.g., 4 turns).

Block 1312 includes, for a given number of turns, evaluating whether theinductance of the device is within a given design criteria (e.g.,inductance range between 50 and 300 nH). In such a scenario, designswithin the design criteria may be nominated. The nominated designs andtheir respective design configuration information are passed along tosubsequent method blocks.

Block 1314 (level-3) includes increasing or otherwise adjusting a traceclearance. The trace clearance may include a distance between adjacentconductors (e.g., petals). As an example, the trace clearance may beinitially selected based on a breakdown voltage of air (e.g., 30 kV/cm).For instance, the initial trace clearance may be selected to be 1 mm.Subsequently, the clearance may be gradually increased to provide anoptimized AC and/or DC resistance value of the inductor. In someembodiments, the AC resistance is proportional to trace clearance. Thatis, increasing the trace clearance increases the AC resistance.

In some embodiments, blocks 1306, 1308, 1310, and 1312 may be consideredas an inductance optimization process 1316.

Block 1318 includes updating the minimum fabrication constraints (e.g.,lithography needs, via diameter, conductor thickness) based on the traceclearance.

Block 1320 includes rechecking that the device inductance is stillwithin the inductance range evaluated in block 1312 (e.g., 50-300 nH).If the device is not within the desired inductance range, the method1300 may include reevaluating the outer diameter of the device (block1306) and/or the number of turns (block 1310) so as to achieve thedesired inductance value.

Block 1322 (level-4) includes adjusting the conductor thickness. Forexample, various thickness values of copper for the upper and lowerconductor may be considered so as to provide a desired maximum amount ofallowable current. For example, the maximum allowable current may begreater than 10 amps and/or greater than 18 amps. Other maximumallowable current values are possible.

Block 1324 includes determining whether device power loss is less than apredetermined value (e.g., 1%, 5%, 10%, or another value). In otherwords, block 1324 may include determining an efficiency of a giveninductor design.

In some embodiments, blocks 1314, 1318, 1320, 1322, and 1324 may beblocks of a petal optimization process 1326.

Block 1328 (level-5a) includes increasing a via diameter so as to reducea resistance of the inductor device. Block 1330 includes updating one ormore fabrication constraints based on the via diameter selected. Thefabrication constraints may include, for example, distance betweenadjacent vias, via position, and a number of vias per petal. Block 1332includes determining whether the via resistance is less than a thresholdvia resistance amount (e.g., a maximum resistance per via). If theresistance is greater than the maximum via resistance, the method 1300may return to block 1328 to increase the via diameter.

Block 1334 (level-5b) includes increasing a number of vias so as toreduce a resistance of the inductor device. Block 1336 includes updatingfabrication constraints based on the number of vias selected. Thefabrication constraints may include, for example, distance betweenadjacent vias, via position, and a number of vias per petal. Block 1338includes determining whether the via resistance is less than a thresholdvia resistance amount (e.g., a maximum resistance per via). If theresistance is greater than the maximum via resistance, the method 1300may return to block 1334 to increase the via diameter. In someembodiments, blocks 1328, 1330, 1332, 1334, 1336, and 1338 may be blocksin a via optimization process 1340.

If blocks 1332 and 1338 indicate that the resistance of the vias of agiven inductor device design is less than the maximum via resistance,the inductor device design may be modeled using finite element analysissoftware (e.g., JMAG). Namely, temperature rise (block 1342), inductance(block 1348), AC resistance (block 1344), and efficiency (block 1346)may be simulated.

Block 1350 is a determination of whether results of the finite elementmodeling analysis pass predetermined requirements for the inductordevice. If the determination indicates that a given design did not passthe predetermined requirements, the method 1300 may return to a priormethod step (e.g., block 1302). In other embodiments, the method 1300may return to

FIG. 14 illustrates a three-dimensional graph 1400 of inductance versusAC resistance and number of turns, according to an example embodiment.In particular, graph 1400 illustrates a multi-objective optimization ofinductance and AC resistance versus the number of turns and varioustrace clearances of a PCB-embedded inductor with 4-oz copper on 2-layerPCB. The inner diameter is varied between 20 mm, 25 mm, and 30 mm.

FIG. 15 illustrates a graph 1500 of inductance and efficiency versusnumber of turns, according to an example embodiment. Namely, graph 1500illustrates example output from a multi-objective optimization ofinductance and estimated efficiency versus the number of turns in aPCB-embedded inductor with 4-oz copper on a 2-layer PCB. The threedifferent plots show various inner diameters of 15 mm (highestinductance, lowest efficiency), 25 mm, and 30 mm (lowest inductance,highest efficiency).

The particular arrangements shown in the Figures should not be viewed aslimiting. It should be understood that other embodiments may includemore or less of each element shown in a given Figure. Further, some ofthe illustrated elements may be combined or omitted. Yet further, anillustrative embodiment may include elements that are not illustrated inthe Figures.

A step or block that represents a processing of information cancorrespond to circuitry that can be configured to perform the specificlogical functions of a herein-described method or technique.Alternatively or additionally, a step or block that represents aprocessing of information can correspond to a module, a segment, or aportion of program code (including related data). The program code caninclude one or more instructions executable by a processor forimplementing specific logical functions or actions in the method ortechnique. The program code and/or related data can be stored on anytype of computer readable medium such as a storage device including adisk, hard drive, or other storage medium.

The computer readable medium can also include non-transitory computerreadable media such as computer-readable media that store data for shortperiods of time like register memory, processor cache, and random accessmemory (RAM). The computer readable media can also includenon-transitory computer readable media that store program code and/ordata for longer periods of time. Thus, the computer readable media mayinclude secondary or persistent long term storage, like read only memory(ROM), optical or magnetic disks, compact-disc read only memory(CD-ROM), for example. The computer readable media can also be any othervolatile or non-volatile storage systems. A computer readable medium canbe considered a computer readable storage medium, for example, or atangible storage device.

While various examples and embodiments have been disclosed, otherexamples and embodiments will be apparent to those skilled in the art.The various disclosed examples and embodiments are for purposes ofillustration and are not intended to be limiting, with the true scopebeing indicated by the following claims.

What is claimed is:
 1. A device comprising: a substrate having an uppersurface and an opposing lower surface; a plurality of upper conductorsdisposed along the upper surface, wherein the plurality of upperconductors is radially disposed about a central axis, wherein each upperconductor comprises a petal shape, wherein a distance between adjacentupper conductors is less than a width of each upper conductor; aplurality of lower conductors disposed along the lower surface, whereinthe plurality of lower conductors is radially disposed about the centralaxis, wherein each lower conductor comprises a petal shape, wherein adistance between adjacent lower conductors is less than a width of eachlower conductor; and a plurality of through-substrate conductorsconnecting respective upper conductors to respective lower conductors soas to form a series electrical connection, wherein the series electricalconnection comprises a toroid configuration.
 2. The device of claim 1,wherein the toroid configuration comprises a plurality of turns about areference circle that is defined along a reference plane between theupper surface and the lower surface.
 3. The device of claim 1, wherein athickness of the substrate is less than about 1.6 millimeters.
 4. Thedevice of claim 1, wherein a diameter of the device is no greater thanabout 60 millimeters.
 5. The device of claim 1, wherein an area of thedevice is less than about 3000 mm².
 6. The device of claim 1, whereinthe toroid configuration comprises a plurality of turns and theplurality of turns is at least one of 8, 13, or
 26. 7. The device ofclaim 1, wherein an inductance of the device is between about 45 and 305nanoHenries.
 8. The device of claim 1, wherein a resistance of thedevice is less than about 2.6 milliOhms at DC and less than about 9milliOhms at 1 MHz.
 9. The device of claim 1, wherein a quality factorof the device is at least
 300. 10. The device of claim 1, furthercomprising a heat sink coupled to at least one of the upper surface orone of the plurality of upper conductors.
 11. A system comprising: adirect current (DC) source input; a switching circuit coupled to the DCsource input; a filtering circuit coupled to the switching circuit,wherein the filtering circuit comprises an inductor, the inductorcomprising: a substrate having an upper surface and an opposing lowersurface; a plurality of upper conductors disposed along the uppersurface, wherein the plurality of upper conductors is radially disposedabout a central axis, wherein each upper conductor comprises a petalshape, wherein a distance between adjacent upper conductors is less thana width of each upper conductor; a plurality of lower conductorsdisposed along the lower surface, wherein the plurality of lowerconductors is radially disposed about the central axis, wherein eachlower conductor comprises a petal shape, wherein a distance betweenadjacent lower conductors is less than a width of each lower conductor;and a plurality of through-substrate conductors connecting respectiveupper conductors to respective lower conductors so as to form a serieselectrical connection, wherein the series electrical connectioncomprises a toroid configuration; and an output circuit coupled to thefiltering circuit.
 12. The system of claim 11, wherein an input currentof the DC source input is at least 17 Amperes.
 13. The system of claim11, wherein the switching circuit comprises a GaN half-bridge inverter,and wherein a switching frequency of the switching circuit is betweenabout 1 MHz and 30 MHz.
 14. The system of claim 11, wherein theswitching circuit provides a pulse-wave modulated (PWM) signal to thefiltering circuit.
 15. The system of claim 11, wherein the system isoperated in a continuous conduction mode, wherein a peak output voltageof the output circuit is at least 30 Volts, and wherein an outputcurrent of the output circuit is at least 9 Amperes.
 16. The system ofclaim 11, wherein the system is operated in a discontinuous conductionmode.
 17. The system of claim 11, wherein, during operation of thesystem, a temperature rise of the inductor is no greater than about 45°C.
 18. The system of claim 11, wherein a total harmonic distortion (THD)of the output circuit is less than about 10%.
 19. A method for designinga PCB-embedded inductor, the method comprising: adjusting an outerdiameter of the inductor and a number of turns of the inductor based onpredetermined physical constraints and predetermined performancespecifications, wherein the inductor comprises: a plurality of upperconductors disposed along an upper surface of a substrate, wherein theplurality of upper conductors is radial disposed about a central axis,wherein each upper conductor comprises a petal shape, wherein a traceclearance between adjacent upper conductors is less than a width of eachupper conductor, wherein the substrate comprises a printed circuit board(PCB); a plurality of lower conductors disposed along a lower surface ofthe substrate, wherein the lower surface is disposed opposite the uppersurface, wherein the plurality of lower conductors is radially disposedabout the central axis, wherein each lower conductor comprises a petalshape, wherein a trace clearance between adjacent lower conductors isless than a width of each lower conductor; and a plurality ofthrough-substrate conductors connect respective upper conductors torespective lower conductors so as to form a series electricalconnection, wherein the series electrical connection comprises a toroidconfiguration, wherein the toroid configuration comprises a plurality ofturns about a reference circle that is defined along a reference planebetween the upper surface and the lower surface; subsequent to adjustingthe outer diameter and the number of turns, determining whether aninductance of the inductor is greater than a threshold inductance value;adjusting a trace clearance, wherein the trace clearance comprises adistance between adjacent conductors; subsequent to adjusting the traceclearance, determining whether an inductance of the inductor is greaterthan a threshold inductance value; adjusting a thickness of at least oneof the upper conductor or the lower conductor; adjusting a diameter ofthe through-substrate conductors; adjusting a number ofthrough-substrate conductors; determining a plurality of deviceperformance metrics, based on the adjusted attributes of the inductor;determining, based on the device performance metrics being above aplurality of device requirements, that the inductor should befabricated; and fabricating the inductor according to the adjustedattributes.
 20. The method of claim 19, wherein determining theplurality of device performance metrics comprises conducting at leastone finite element analysis.